Co is a prefix, a syllable placed before a word. The word prefix itself is made up of the prefix pre (meaning before) and the word fix (meaning attach).

The prefix co (and its alternative forms con, com, col and cor, depending on which letter follows the prefix) has the meaning with, together with.

A prefix is usually not followed by a hyphen. Some examples:
Afterthought, antedate, biannual, collaborate, commemorate, confederation, displace, ensure, illegal, indirect, overuse, posttraumatic, prepaid, replace, submarine, underestimate, uninterested.

So you are right in leaving out the hyphen in words such as cooperate, collaborate and coordinate.

A grandfather is helping his grandson with a toy car

However, in some cases a hyphen is to prefer, since otherwise the spelling might suggest a different pronunciation:

If you write co-opt without a hyphen (coopt), it looks as if it could be pronounced with a vowel as in too, and re-edit, when written reedit might sound like read it. The same pronunciation issue would apply to, for example, re-enter, re-establish, and re-examine.

Some words with the prefix re- have two versions, one with and one without a hyphen:

When you re-sign a document, you sign it again, but when you resign, you quit a job.
To re-cover means to cover again, while recover is to get better, regain your health.
When you re-store goods, you put them back in store again, but to restore something means to reconstruct or bring back to a former state.

You can read more about re here.

And, since we are dealing with English, we have to accept inconsistencies like the following:

Anti-clockwise, anti-hero and anti-Semitism but antirust, antiseptic and antisocial.
Vice President of a state; vice president or vice-president in business.
Viceroy but vice-chancellor.

A last note: Instead of writing cooperate, you could consider writing collaborate. There is a difference between the two!